\(\int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx\) [378]

Optimal result

Integrand size = 25, antiderivative size = 32 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=-\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^n \log (F)\right ) \log ^4(F)}{d n} \]

[Out]

-F^a/((d*x+c)^n)^4*Ei(5,-b*(d*x+c)^n*ln(F))/d/n

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2250} \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=-\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-b (c+d x)^n \log (F)\right )}{d n} \]

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 4*n),x]

[Out]

-((b^4*F^a*Gamma[-4, -(b*(c + d*x)^n*Log[F])]*Log[F]^4)/(d*n))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^n \log (F)\right ) \log ^4(F)}{d n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=-\frac {b^4 F^a \Gamma \left (-4,-b (c+d x)^n \log (F)\right ) \log ^4(F)}{d n} \]

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^(-1 - 4*n),x]

[Out]

-((b^4*F^a*Gamma[-4, -(b*(c + d*x)^n*Log[F])]*Log[F]^4)/(d*n))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(34)=68\).

Time = 0.64 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.53

method	result	size
risch	\(-\frac {F^{a} \left (\operatorname {Ei}_{1}\left (-b \left (d x +c \right )^{n} \ln \left (F \right )\right ) \ln \left (F \right )^{4} b^{4} \left (d x +c \right )^{4 n}+F^{b \left (d x +c \right )^{n}} \ln \left (F \right )^{3} b^{3} \left (d x +c \right )^{3 n}+F^{b \left (d x +c \right )^{n}} \ln \left (F \right )^{2} b^{2} \left (d x +c \right )^{2 n}+2 F^{b \left (d x +c \right )^{n}} \ln \left (F \right ) b \left (d x +c \right )^{n}+6 F^{b \left (d x +c \right )^{n}}\right ) \left (d x +c \right )^{-4 n}}{24 n d}\)	\(145\)

[In]

int(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x,method=_RETURNVERBOSE)

[Out]

-1/24/n/d*F^a*(Ei(1,-b*(d*x+c)^n*ln(F))*ln(F)^4*b^4*((d*x+c)^n)^4+F^(b*(d*x+c)^n)*ln(F)^3*b^3*((d*x+c)^n)^3+F^
(b*(d*x+c)^n)*ln(F)^2*b^2*((d*x+c)^n)^2+2*F^(b*(d*x+c)^n)*ln(F)*b*(d*x+c)^n+6*F^(b*(d*x+c)^n))/((d*x+c)^n)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (36) = 72\).

Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.72 \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=\frac {{\left (d x + c\right )}^{4 \, n} F^{a} b^{4} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{4} - {\left ({\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} + {\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} + 2 \, {\left (d x + c\right )}^{n} b \log \left (F\right ) + 6\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{24 \, {\left (d x + c\right )}^{4 \, n} d n} \]

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="fricas")

[Out]

1/24*((d*x + c)^(4*n)*F^a*b^4*Ei((d*x + c)^n*b*log(F))*log(F)^4 - ((d*x + c)^(3*n)*b^3*log(F)^3 + (d*x + c)^(2
*n)*b^2*log(F)^2 + 2*(d*x + c)^n*b*log(F) + 6)*e^((d*x + c)^n*b*log(F) + a*log(F)))/((d*x + c)^(4*n)*d*n)

Sympy [F]

\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=\int F^{a + b \left (c + d x\right )^{n}} \left (c + d x\right )^{- 4 n - 1}\, dx \]

[In]

integrate(F**(a+b*(d*x+c)**n)*(d*x+c)**(-1-4*n),x)

[Out]

Integral(F**(a + b*(c + d*x)**n)*(c + d*x)**(-4*n - 1), x)

Maxima [F]

\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=\int { {\left (d x + c\right )}^{-4 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(-4*n - 1)*F^((d*x + c)^n*b + a), x)

Giac [F]

\[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=\int { {\left (d x + c\right )}^{-4 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a} \,d x } \]

[In]

integrate(F^(a+b*(d*x+c)^n)*(d*x+c)^(-1-4*n),x, algorithm="giac")

[Out]

integrate((d*x + c)^(-4*n - 1)*F^((d*x + c)^n*b + a), x)

Mupad [F(-1)]

Timed out. \[ \int F^{a+b (c+d x)^n} (c+d x)^{-1-4 n} \, dx=\int \frac {F^{a+b\,{\left (c+d\,x\right )}^n}}{{\left (c+d\,x\right )}^{4\,n+1}} \,d x \]

[In]

int(F^(a + b*(c + d*x)^n)/(c + d*x)^(4*n + 1),x)

[Out]

int(F^(a + b*(c + d*x)^n)/(c + d*x)^(4*n + 1), x)